, 2012

Let $A_1S$, $B_1T$ and $C_1U$ be the altitudes of $A_1B_1C_1$. The circle $k$ with the diameter $A_1H_1$ contains the points $A_1$, $H_1$, $T$, $U$, $P$ and $Q$. Let $AA_1$ intersect the incircle of $ABC$ for the second time at $V$. Assume that $\angle B \ge \angle C$.

Observe that $\angle C_1A_1V = \angle TA_1P$, $\angle C_1B_1A_1 = \angle C_1A_1B = \angle TA_1Q$, and $\angle VA_1B_1 = \angle PA_1U$. Considering the chords subtending these angles in the circle $k$ and in the incircle of the triangle $ABC$ we conclude that the cyclic quadrilaterals $PTQU$ and $VC_1A_1B_1$ are similar.

Let $M$ and $N$ be the midpoints of the line segments $B_1C_1$ and $A_1H_1$. Notice that
$$\angle MTN = 180^\circ - \angle C_1TM - \angle NTA_1 = 180^\circ - \angle TC_1M - \angle NA_1T = 90^\circ$$
as $M$ and $N$ are the midpoints of the hypotenuses of the right triangles $C_1TB_1$ and $TH_1A_1$, respectively.

Therefore $MT$ and, similarly, $MU$, are tangent to $k$. It follows that the pentagons $PMTQU$ and $VAC_1A_1B_1$ are similar. Since the points $A_1$, $V$, $A$ lie on a line, so do the corresponding points $Q$, $P$, $M$.